WASHINGTON—Rear Adm. Ronny Jackson dropped his bid Thursday to lead the Veterans Affairs Department, ending a Senate inquiry into his background but still facing a likely military review into allegations about his conduct while serving as the president’s personal physician.

Dr. Jackson’s decision to pull out of the running for the VA job came hours after the release of a two-page summary of allegations compiled by the Senate Veterans’ Affairs Committee’s Democratic staff.

In a statement Thursday morning announcing his decision to withdraw, Dr. Jackson called the allegations “completely false and fabricated.” President Donald Trump called him a “great man” who had been treated unfairly.

Dr. Jackson’s wife phoned him on Thursday and told him that reporters were camped out at their home, a disturbing development that fed his frustration over the confirmation proceedings, a White House official said.

The Democratic summary, assembled from interviews with about two-dozen current and former colleagues of Dr. Jackson, included allegations that he had dispensed prescription drugs without paperwork and wrecked a car while drunk. It repeated an allegation from Sen. Jon Tester, the top Democratic on the committee, that Dr. Jackson’s nickname was “The Candyman” because he so freely gave out prescription medication.

U.S. defense officials said that the Navy only learned of the accusations during Dr. Jackson’s bid to become a member of the cabinet. While at the White House, Dr. Jackson was under the supervision of the White House Military Office and largely outside the traditional Navy chain of command, the officials said.

Either the Defense Department Office of Inspector General or the Navy Office of Inspector General likely would be assigned to undertake an investigation, the defense officials said. That decision could be determined by which office receives any complaints.

Anyone, including civilians, can ask military inspector general offices to look into complaints, the officials added.

The Navy investigated Dr. Jackson in 2012 when the Navy Bureau of Medicine and Surgery looked into complaints that then-Capt. Jackson and a fellow Navy captain had created an unhealthy environment at the White House office they led, Navy documents show. Despite the concerns, both Dr. Jackson and his colleague remained on the job, defense officials said.

Dr. Jackson returned to work at the White House on Thursday. White House press secretary Sarah Sanders indicated he would remain in his current posting. The White House is intent on clearing Dr. Jackson’s name.

But one administration official said the White House had reservations about commissioning an inspector general report to investigate, saying that “even if they found one small little thing, it would give credibility” to the larger set of allegations against Dr. Jackson.

With Dr. Jackson’s withdrawal, speculation turned to whom Mr. Trump would nominate next to lead a 370,000-person agency with a legacy of troubled management.

People close to the White House said that two likely candidates are former Republican Rep. Jeff Miller (R., Fla.) and Ike Perlmutter, the chairman and former CEO of Walt Disney Co.’s Marvel Entertainment unit.

Mr. Miller chaired the House Veterans Affairs Committee before leaving Congress in 2017. He endorsed Mr. Trump during the campaign.

In an interview last year, Mr. Miller said of the president: “I believe he is widely accepted as the person who can change the VA for the next generation.” Mr. Miller didn’t respond to a request for comment.

After leaving Congress in 2017, Mr. Miller signed on with the firm McDermott Will & Emery LLP. He is a registered lobbyist for at least five clients with issues before the Department of Veterans Affairs, according to lobbying reports.

He represents billionaire Steven A. Cohen, who bankrolled a network of private veterans clinics and has been asking Congress and Veterans Affairs to increase federal funding of mental health care for veterans and expand combat trauma research, according to lobbying reports.

Mr. Perlmutter’s wife, Laura Perlmutter, served on Mr. Trump’s inaugural committee and gave $1 million to Mr. Trump’s foundation in 2016, according to public records. He has often dined with Mr. Trump at his Florida resort, Mar-a-Lago, and flown with him on Air Force One.

Mr. Trump offered few hints about the next nominee. In a Fox News interview Thursday, he would only say he has a candidate in mind—someone with “political capability.”

The White House isn’t expected to announce the next nominee in the coming days, as it seeks to more thoroughly vet the new candidates. “We’ll do our due diligence this time and try to do a little bit of vetting before announcing,” one administration official said.

The nomination drama Thursday also threw a spotlight on Mr. Tester and his decision to air the allegations during the Senate confirmation proceedings.

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“That was not his finest day in the Senate,” Sen. Pat Roberts (R., Kan.) said of Mr. Tester.

Mr. Trump suggested on Fox that voters in Mr. Tester’s home state of Montana wouldn’t forgive the senator. Mr. Tester is running for re-election this fall in a state Mr. Trump won in 2016 with 56% of the vote.

“I think Jon Tester has to have a big price to pay in Montana,” the president said.

Later, in an appearance in the Rose Garden, Mr. Trump said that Dr. Jackson is “a great man” who had been “treated very, very unfairly.”

Democrats noted that Mr. Tester had presented the information with the caveat that he was still investigating the veracity of some accusations.

“The volume of complaints and the number of people who came forward—both of those things are very significant,” said Sen. Dick Durbin (D., Ill.). “This is not just one idle rumor.”

Mr. Tester said Thursday that he was “absolutely” comfortable with his decision to air the allegations publicly.

“We made clear what that document was from the get-go,” Mr. Tester said. “There was no claim ever that this was all a fact.”